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5 Sampling Distributions for Means.pdf

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University of Guelph
STAT 2080
John Walsh

ST2080 Ali 51Sampling Distributions for MeansEverything we have learned so far in this course has set us up for understanding that Characteristics in a population have a distribution Randomly selecting units from the population help us learn about that distribution Sample statistics can be used to estimate a population level parameters A sample statistic has its own probability distribution because different samples of the same size can give rise to different values of the statistic Lets study the last two points by through the following example Eg Suppose there are 8 students in the population from which we are interested in estimating the average spare cash that they have on them on a given day Although unknown to us assume the amount of spare cash each has on the given day is as followsStudent 1 2 3 4 5 6 7 8 x4 2 1 4 7 7 8 7 i The true average amount of spare cash for this population is However once units are selected for the sample we can record the desired characteristic x from each sample unit and use the sample iaverage statistic to estimate the populationlevel mean parameterFor a SRS of size 4 out of a population of size 8 there are 70 possible samples and the sample average spare cash may be the same for some of these samples analogous to how there are 36 possible outcomes from tossing two dice some of which give the same sum of the top facesSois a random variable itself whose value depends on which sample from the 70 possible samples is actually drawn from the population 87ST2080 AliStudents in Spare CashSampleSample data xi1 1 2 3 4 4 2 1 4 275 2 1 2 3 5 4 2 1 7 350 3 1 2 3 6 4 2 1 7 350 70 5 6 7 8 7 7 8 7 725Q What is the sampling distribution ofie What are the possible values ofand with what probability would we observe those values275350375400425450475500525550575600625650725Freq 16237 46 126473 26 1 PWe can compute the mean of the sample mean exactly from first principleHowever this requires us to actually enumerate all possible samples of size 4 compute the sample mean for each possible sample and find the true sampling distribution WE NEED A SHORTCUT 88
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